Wait. What does work for you?
For example:
If I enter "opera back" in the adress bar, without the quotes, not a single suggestion pops up.
However, if I type that in the search bar on google homepage, I do get suggestions like "opera background image". This is exactly what I want.

If I type something in Opera's address bar, I see suggestions from Google. Of course that requires Google still set as default search engine Also note that some privacy software will block access to suggestions.

If I type something in Opera's address bar, I see suggestions from Google. Of course that requires Google still set as default search engine Also note that some privacy software will block access to suggestions.

Does a search via adress bar really leads you to the same suggestions as if you had typed them in the search bar on google homepage? Would you mind show some screenshots? Would appreciate that.

Well, barring that the address bar is limited to 5 suggestions while the Google homepage shows 10. Opera only showed the first 5 from the ones I saw on Google. (Actually, I wasn't sure they'd be the same when you asked - Google does try to tailor results based on your history and location and so on. Though your "Opera back" example is unlikely to care about location, as opposed to typing something like "mexican restaurant".)